Fuller invented the Geodesic Dome in the late 1940s to demonstrate some
ideas about housing and ``energetic-synergetic geometry'' which he had
developed during WWII. This invention built on his two decade old quest
to improve the housing of humanity. It represents a brilliant
demonstration of his synergetics principles; and in the right
circumstances it could solve some of the pressing housing problems of
today (a housing crisis which Fuller predicted back in 1927).

[From Robert T. Bowers' paper on Domes last posted to
GEODESIC in 1989.]

A geodesic dome is a type of structure shaped like a piece of a sphere
or a ball. This structure is comprised of a complex network of triangles
that form a roughly spherical surface. The more complex the network of
triangles, the more closely the dome approximates the shape of a true
sphere [sic].

By using triangles of various sizes, a sphere can be symmetrically
divided by thirty-one great circles. A great circle is the largest
circle that can be drawn around a sphere, like the lines of latitude
[ED: he means longitude]
around the earth, or the equator. Each of these lines divide the sphere
into two halves, hence the term geodesic, which is from the Latin
meaning ``earth dividing.''

[From Mitch Amiano]

The dome is a structure with the highest ratio of enclosed area to
external surface area, and in which all structural members are equal
contributors to the whole. There are many sizes of triangles in a
geodesic [ED: dome], depending on the frequency of subdivision of the
underlying spherical polyhedron. The cross section of a geodesic [ED:
dome] approximates a great-circle line.

Do domes really weigh less than their component materials?

[From Pat Salsbury]

Well, the structures weigh less when completed because of the air-mass
inside the dome. When it's heated warmer than the outside air, it has a
net lifting effect (like a hot-air balloon).

This is almost unnoticeable in smaller structures, like houses, but, as
with other things about geodesics, being as they're based upon spheres,
the effect increases geometrically with size. So you'd be able to notice
it in a sports stadium, and a sphere more than a half mile in diameter
would be able to float in the air with only a 1 degree F difference in
temperature!

What about underground concrete domes?

[From Randy Burns.]

Underground concrete domes are rather interesting

1) They can use chemical sealing and landscaping to avoid leakage
problems associated with wooden domes.

2) They are extremely strong. Britz [see
Dome References for more on Britz]
has obtained extremely low insurance rates on his structures. The
insurance company tested one building by driving a D8 Caterpillar tractor
on top of the house!

3) There's little hassle involved in dealing with materials that were
really standardized for use building boxes. The only specialized tools
are the forms, everything else can easily be used off the shelf.

4) They can be quite aesthetic. Britz has shown that you can build
developments where the houses can't really see each other.

5) They are cheap and easy to heat, cheap enough that you can build a
much larger structure than you might using conventional housing and use
standard room divider technology to split the thing up into room.

What are geotangent domes?

[Keyed in by Patrick G. Salsbury.]

The following is quoted from ``Scientific American'' in the September 1989
issue. (Pages 102-104)

Surpassing the Buck
(Geometry decrees a new dome)

``I started with the universe--as an organization of energy systems of
which all our experiences and possible experiences are only local
instances. I could have ended up with a pair of flying slippers.''
-R. Buckminster Fuller

Buckminster Fuller never did design a pair of flying slippers. Yet he
became famous for an invention that seemed almost magical: the geodesic
dome, an assemblage of triangular trusses that grows stronger as it grows
larger. Some dispute that Fuller originated the geodesic dome; in
Science a la Mode, physicist and author Tony Rothman argues that the
Carl Zeiss Optical Company built and patented the first geodesic dome in
Germany during the 1920's. Nevertheless, in the wake of Fuller's 1954
patent, thousands of domes sprung up as homes and civic centers--even as
caps on oil-storage tanks. Moreover, in a spirit that Fuller would have
heartily applauded, hundreds of inventors have tinkered with dome
designs, looking for improved versions. Now one has found a way to
design a completely different sort of dome.

In May, J. Craig Yacoe, a retired engineer, won patent number 4,825,602
for a ``geotangent dome,'' made up of pentagons and hexagons, that
promises to be more versatile that its geodesic predecessor. Since
Fuller's dome is based on a sphere, cutting it anywhere but precisely
along its equator means that the triangles at the bottom will tilt
inward or outward. In contrast, Yacoe's dome, which has a circular base,
follows the curve of an ellipsoid. Builders can consequently pick the
dimensions they need, Yacoe Says. And his design ensures that the
polygons at the base of his dome always meet the ground at right angles,
making it easier to build than a geodesic dome. He hopes these features
will prove a winning combination.

Although Fuller predicted that a million domes would be built by the
mid-1980's, the number is closer to 50,000. Domes are nonetheless still
going up in surprising places. A 265-foot-wide geodesic dome is part of
a new pavilion at Walt Disney World's Epcot Center in Florida. A bright
blue 360-foot-high dome houses a shopping center in downtown Ankara,
Turkey. Stockholm, Sweden, boasts a 280-foot-high dome enclosing a new
civic center.

Dome design is governed by some basic principles. A sphere can be
covered with precisely 20 equilateral triangles; for a geodesic dome,
those triangles are carved into smaller ones of different sizes. But to
cover a sphere or ellipsoid with various sizes of pentagons and hexagons
required another technique, Yacoe says.

Yacoe eventually realized that he could build a dome of polygonal panels
guided by the principle that one point on each side of every panel had
to be tangent to (or touch) an imaginary circumscribed dome. With the
assistance of William E. Davis, a retired mathematician, he set out to
describe the problem mathematically.

They began with a ring of at least six congruent pentagons wrapped
around the equator of an imaginary ellipse. The task: find the lengths
of the sides and the interior angles of the polygons that form the next
ring.

To do so for an ellipsoidal dome, they imagined inscribing an ellipse
inside each polygon. Each ellipse touched another at one point; at these
points, the sides of the polygons would also be tangent to a
circumscribed ellipsoid. But where, precisely, should the points be
located? Yacoe and Davis guessed, then plugged the numbers into
equations that describe ellipses and intersecting planes. Aided by a
personal computer, they methodically tested many guesses until the
equations balanced. Using the tangent points, Yacoe and Davis could then
calculate the dimensions and interior angles of the corresponding
polygons and so build the next ring of the dome.

After receiving the patent, Yacoe promptly set up a consulting firm to
license his patents. He says dome-home builders have shown considerable
interest, as has Spitz, Inc., a maker of planetariums located near Yacoe
in Chadds Ford, Pa. Yacoe has also proposed that the National
Aeronautics and Space Administration consider a geotangent structure as
part of a space station. -E.C.

What are the advantages (and disadvantages) of Dome Life?

asemon@esu.edu (Alan Semon) writes:
>I was once interested in the idea of living in a geodesic dome home and,
>to the best of my recollection, these are some of the advantages:
>
>1. Heating and cooling the home become more efficient due to the fact
>that there are fewer (even no) corners where heat may be trapped. The
>overall air flow in a dome is substantially better than in a
>conventionally constructed home (straight walls and such).
>
...and there is less surface area per square foot of living space = less
heat loss.
>2. Many dome home designs allow the option of using larger lumber for
>the dome. 2x6's or 2x8's instead of the usual 2x4's, although this is
>an option in ANY home, it seems to be more commonly done in dome home
>construction.
>
Although for many areas of the US, there is no financial advantage to
using 2x6 construction. A dome with R-14 throughout can outperform a
well insulated conventional house of comparable S/F.
>3. For those solar minded people, the placement of the solar collectors
>on the ``roof'' is less critical due to the curved nature of the top of
>the structure.
>
>4. The inherent strength of the dome makes it suitable for either
>earth-bermed or even earth covered construction techniques. In the case
>of more common construction techniques, the structural members'
>dimensions usually need to be completely reworked in order to carry the
>extra weight.
>
>5. Hell, they _LOOK_ pretty neat! This might be a problem in certain
>areas which one of those laws which say that all homes in an area _MUST_
>conform to certain guidelines concerning their architecture (bummer,
>huh? :-)).
-jg

[Based in part on a Brewer Eddy post]

The curved walls in a dome require either custom furnishings,
100% prefab design, or an ``open spaces'' approach. Each of these
would be an advantage or disadvantage in one person's eyes or
another's.

Mass producing domes is easy, greatly reduces the cost and could solve
many of the housing shortage problems worldwide (especially emergency
housing needs).

How to use solar panels in domes? [Kerri Brochard]

[From Tom Dosemagen]

I have a dome and tried to find solar panels to be installed on the
dome. I had no luck finding such a beast so I installed 320 square feet
of panels on the ground close to the dome and ran all connections under
ground into the basement. I live in south central Wisconsin and my
experience with solar is not the greatest. My system works fine, but in
order for the system to work the sun has to shine. That doesn't happen
a lot here until late February or early March. My advice to people in our
part of country is to take the money you were going to spend on solar
and invest it. Then take your interest money and pay for conventional
heat. My dome is 44 feet in diameter and with a 90% efficient furnace
and my total heating bill for one season is right around $350.00. My
exterior walls are framed with 2x6's. With thicker dome walls I'm sure
that I could lower my heating costs by quite a bit.

Dome Theory

[From Kirby Urner.]

The edges of a geodesic dome are not all the same length. The
angstrom measurements between neighboring carbon atoms in a fullerene
are likewise not equal.

Domes come in three Classes (I, II and III). The classification system
has to do with laying an equilateral triangle down on a grid of smaller
equilateral triangles, lining up corners with corners -- either aligning
the triangle with the grid (I), turning it 90 degrees to bisect grid
triangles (II), or rotating it discretely to have it cut skewly across
the grid (III).

20 of these triangles make an icosahedron which is then placed within a
circumscribing sphere. The vertexes of the triangles' internal points,
defined by the grid pattern, define radii with the circumscribing
sphere's center. By pushing each vertex further out along the segments
so defined, until each is made equidistant from the center, an
omnitriangulated geodesic sphere is formed (orthonormal projection I
think cartographers call this). Again, resulting surface edge lengths
are not all the same length. The resulting mesh will always contain 12
sets of 5 triangles organized into pentagons, the rest into hexagons.

The Class I version of the algorithm above always creates 20F^2 surface
facets where F=1 gives the icosahedron itself. The external point
population will be 10F^2+2. Since points plus facets = edges plus 2
(Euler), you will get 30F^2 edges. F is what Fuller called the
Frequency of the geodesic sphere and, in the Class I case, corresponds
to the number of grid intervals along any one of the 20 triangle edges.

Note: ``buckyballs'' in the sense of ``fullerenes'' are not
omnitriangulated (the edges internal to the 12 pentagons and n hexagons
have been removed) and come in infinitely more varieties than the above
algorithm allows. The above algorithm is limited to generating point
groups with icosahedral symmetry -- a minority of the fullerenes are
symmetrical in this way, although C60, the most prevalent, is a derivative
of the Class I structure.

[From Ben Williams]
Andrew Norris writes:
>1/ Given a dodecahedron with the edges of length unity, what is
> the radius of the sphere that would enclose this body?
>
>2/ For the above case, construct each pentagon out of triangles.
> What are the angles required so that new center-node of the
> pentagon just touches the enclosing sphere?

This is just a 2 frequency (what-is-referred-to-in-Domebook II-as)
triacon geodesic sphere. Funny you should mention that: Back in June
when I first discovered this newsgroup, I got reinterested in my old
hobby of building mathematical models (and R B Fuller as well). So I
went through the laborious process of calculating the strut lengths to
build a 2v triacon sphere (what you just described above) out of
toothpicks. I have it hanging up over my monitor right now. I wish I
could show how I used geometry and such to figure all the necessary
lengths out. What I do is start out with a drawing of a dodecahedron
projected onto a plane -- if it is oriented correctly, you will get a
2-d figure that you can use to deduce the information you want from it.
(To get this figure, think of a dodecahedron made out of struts (such as
toothpicks) standing on one of its edges on a sheet of paper out in the
sun with the sun directly overhead. The shadow on the paper will be
this figure.) These are the lengths I derived

E = length of edge of dodecahedron
Distance of edge of dodecahedron from center:

Er = ( (3 + sqrt(5))/4 ) * E

1/2 distance between non-adjacent vertices of face of dodecahedron:

b = ( (sqrt(5)+1)/4 ) * E

given a face of dodecahedron, distance between vertex and opposite
edge:

h = ( ( sqrt(5 + 2*sqrt(5)) ) / 2 ) * E

distance from center of dodecahedron to one of its vertices (your
question 1):

R = sqrt((9 + 3*sqrt(5))/8) * E

given a face of dodecahedron, distance from its center to an edge:

l = b/h * Er

distance from center of face of dodecahedron to center of
dodecahedron:

m = Er/h * Er

given face of dodecahedron, distance from center to vertex:

t = h-l

length of one of those struts going from a vertex of dodecahedron up
to point above center of face but on the enclosing sphere:

S = sqrt(t^2 + (R-m)^2)

Now, to derive the angles of one of those triangles whose side lengths I
have just determined, you would need to do this:

A1 = 2 * arcsin ((E/2)/S)

This is the angle of the top corners of the 5 triangles which are arched
above one of the faces of the dodecahedron. My calculator gives me this
angle in degrees: 67.66866319 Notice it is slightly less than the 72
degrees it would be if they were flat on the face of the dodecahedron.
Now the other two angles of each of the triangles are simply derived
via:

A2 and A3 = (180 - A1) / 2

I get a value of 56.1656684 degrees for these two angles.

What are the basics of Spherical Trigonometry?

On Sat, 18 Dec 1993 03:11:53 GMT <scimatec5@UOFT02.UTOLEDO.EDU> said:
>Hey all,
> A while back I asked about calculating chord factors. I found the
>equation that without which I don't think I could have done it (by the way I
>was successful)-- it's a formula for calculating w/any spherical right
>triangle. The formula is sin a = sin A * sin c.
> A
> / |
> c / |b
> / |
> / |
> B--a--C
>I'm sure you're all familiar w/it, but is there any other equation that would
>be just as helpful.
This is by Napier's rules. Here is Napier's circle: c-c
A-c B-c
b a

where -c means the complement (or 90 degrees - (minus) the arclength measure).
A, B are angles, C is the right angle and a, b, c are the sides opposite A, B,
and C, respectively. There are two rules:

Rule 1:

The sine of any unknown part is equal to the product of the
cosines of the two known opposite parts. Or sin = cos * cos of the
OPPOSITE parts.

Rule 2:

The sine of any unknown part is equal to the product of the
tangents of its two known adjacent parts. Or sin = tan * tan of the
ADJACENT parts.

Your formula is the same because ``c-c''=90-c and sin(90-c)=cos(c).
Examples: sin(b)=tan(A-c)tan(a) or sin(b)=cos(c-c)cos(B-c).

>
> Steve Mather
Chris Fearnley

How to tesselate a sphere?

[From an old comp.graphics FAQ, posted by Christopher McRae 14 Apr
1993.]
One simple way is to do recursive subdivision into triangles. The base
of the recursion is an octahedron, and then each level divides each
triangle into four smaller ones. Jon Leech leech@cs.unc.edu has
posted a nice routine called sphere.c that generates the coordinates.
It's available for FTP on ftp.ee.lbl.gov and
weedeater.math.yale.edu.

Chord Factors - the nitty gritty.

First choose a tessellation of the sphere (icosa, octa, tetra, elliptical
or really just about anything. Second use geometry and spherical trig
to determine the surface arclengths for the specific tessellation. Third
observe that in any circle a central angle cuts off an arc with the same
exact measure. Next, calculate the chord factors: cf = 2sin(theta/2),
where theta is the central angle. Finally, multiply each chord factor
by the radius of your dome.

Several dome books use the term ``alternate'' to refer to Class I domes
(actually it seems Joe Clinton in his paper on domes has determined
several methods for class I subdivisions - his method I is the
``alternate'' form). The other popular subdivisioning scheme is based on
the rhombic triacontrahedron and is called ``triacon.''

[From Steve Mather]

Hey all, I have some questions to ask about the trigonometry behind
geodesic domes. Remarkably, I've understood what I've encountered so
far, and am well on my way to calculating the the chord factors for a 5v
icosa alternate (Why? when I can look it up in a book? Well, I figured
I'd prove to myself I can.) I've been able to find those along the
direct projection from the icosahedron (are 0.198147431 w/central angle
of 11.3716678 degrees, 0.231597598 w/central angle of 13.29940137, and
0.245346417 w/central angle of 14.09281254 accurate beginnings for the
outside?

[A big thanks to Steve for calculating and typing in all this for
us!!! I'm not certain about the results, but he did such a careful job
that I suspect they are correct. I'm sure someone will check this more
carefully. Please let me know of any problems.]

The letters begin at the bottom of the horizontal edges to the triangle,
from ``a'' to whatever letter (depending upon the frequency --``a'' is the
very bottom, as well as the sides.) The numbers are the chord factors.

The way I calculated my factors was like this:

I took the frequency (f) and divided the degree of the
central angle of that frequency. I then multiplied this
number times the number of rows down the row of lines are
(check figure.) I took the sine of this number and
multiplied it times the sine of the face angle (the angle
between the great circles) to find the sine of half of the
angle across the row (whew-- is this making any sense? =)
I then multiply this angle times two and divide by the
number of rows down (check second sentence and figure.)

This gives me the angle of the geodesic I want. I then
take these numbers and divide by two, take the sine and
multiply by two, to find the chord factor.
These chord factors are multiplied times the radius to
get their lengths.

Here are the equations used:

f= frequency
n= number of rows
A= face angle
All numbers are in degrees
2 sin^-1((sin((63.43494885/f)*n))*sinA))

(the extra ")" shouldn't be there.
sorry, my computer's acting up, and for some reason I can't delete
it.)
That was the equation for getting the geodesic. The chord factors
are done from those by the following equation:

If there is any one Frequently Asked Question online in the 'Fuller
School' (an unsupervised collection of mailing lists, Web pages and
other online forums relating to R. Buckminster Fuller ) it is ``How do I
build a geodesic dome?''

Fuller did not invent the geodesic dome. It was invented by Walter
Bauersfeld of the Zeiss Optical Works in Jena, Germany in 1922, and the
first use of it was as a planetarium on the roof of Zeiss that year.

However, Fuller was awarded several patents for the dome. Among them are
US patent #2682235 (1954), US patent #288171 (1959), US patent
#2905113 (1959), US patent #2914074 (1959), etc. Moreover,
Fuller was the one who popularized the technology and pointed out the
dome's advantages and the reasons for its great strength.

Since Bauersfeld conceived of his structure merely as a planetarium
projector (a truly impressive feat) whereas Fuller had a more
comprehensive vision of the geometrical and engineering significance
of the dome. Which man should win history's designation as "The
inventor of the dome"? I'll let the historians and the pundits
debate that one.

Dome Vendors

The list below has been enhanced by contributions from Joe Moore, Gary
Lawrence Murphy, Garnet MacPhee, Robert Holder, and Matthew V. J. Whalen.
This list is alphabetical. AT&T's
AnyWho service provides a way to check for current
information about any company including these vendors.

The two Domebooks -- Domebook, and Domebook Two -- were published in the
early 1970s and are now out of print. They were written in much the
same fashion as the Whole Earth Catalog, with readers sending in
descriptions of their experiences and problems with domes, and the
book's staff arranging the pieces, working in photographs and line
drawings, etc. They are still often available in libraries, or though
university interlibrary-loan. The full citation is:

There was also a book edited by John Prenis (or Prentis, maybe) called
The Dome Builders Handbook (Philadelphia: Running Dog Press, ca. 1975).
There were two editions of this, as well.

Lloyd Kahn has published three other books that contain information on
dome-building: Shelter (which described a wide variety of self-built
homes from all over the world), Shelter II (ISBN 0-394-50219), and a
pamphlet called Refried Domes (Bolinas: Shelter Publications, 1990)
(ISBN 0-936070-10-2). The latter contains the chord factors and angles
for 8-frequency domes (critical information, and unavailable anywhere
else as far as I can tell), suggestions about construction, and some
second thoughts about domes as permanent shelter. If these books are
not in your bookstore, you can order them directly from

If you're interested in learning something about the history of domes in
the counterculture, look up Charles Jencks and William Chaitkin,
Architecture Today (New York: Harry Abrams, 1982). The magazine
Futurist has also published a couple articles on domes in the last
couple years.

Another book to look for Steve Baer, Dome Cookbook (Lama Publications,
1968); as I recall, it has tables for computing strut lengths and some
useful information about dome construction.

National Association of Dome Home Manufacturers
2506 Gross Point Road
Evanston, Illinois 60201

[From Gary Lawrence Murphy and Chris McRae]

Hugh Kenner's ``Geodesic Math and How to Use It'' Berkeley : University of
California Press, c1976. xi, 172 p. : ill. ; 22 cm. (ISBN
0-520-02924-0) This is an excellent book for the hobbyist model
builder, but also shows geometric derivations for a number of
approaches to carving up the surface of a sphere into the
smallest practical number of different shaped parts, which is
the key matter in dome fabrication. The book also discusses
tensegrity designs, although I believe Hugh has since release a
volume devoted to tensegrity. For those without calculators
:-), the appendix of the book lists the dome-vertex values for
many practical frequencies in the basic polyhedral forms.

[From Alex Soojung-Kim Pang, 25 Feb 1992]

A technically useful book is Edward Popko, Geodesics (Detroit: U.
Detroit press, 1968). It has lots of photographs, plans for domes made
from a whole host of materials, different assembly methods, etc..

Another alternative is concrete, earth sheltered domes. These aren't
necessarily geodesic structures. Still, they may well be closer to
nearing widespread commercial use than most geodesic structures.

Three Companies involved in this:

Utopia Designs, Eugene OR (founded by Norm Waterbury)
These are definitely oriented to the do-it-yourselfer. They specialize in
selling forms and blueprints for domes build using inflatable forms.
EarthShips, Eugene, OR
This company was founded by Richard Britz, author of the Edible City
Resource Manual. They specialize in turnkey structures and are more
oriented towards larger developments. Britz does _wonderful_
architectural drawings.
Monolithic Structures, Idaho and Stockton CA
These folks are primarily involved in building _large_ structures, mainly
industrial buildings and grain silo's.

[More concrete companies from Russell Miller. 1994]

The following three companies deal with concrete shell domes, some of
which are geodesic, but none of which are specifically ``Earth
Sheltered.''

Tapitallee is a rainforest retreat centre who run seminars on
alternative technologies etc as well as personal growth type stuff. I
gather some of their buildings are domes. I'm thinking of spending some
time there.